Single Step Stress Relaxation Research Articles

Characterizing the hyper-viscoelastic behavior of adhesive films

Abstract In this study, the hyper-viscoelastic behavior of adhesive films was characterized. A constitutive model was developed by combining the Mooney–Rivlin hyperelastic model and a viscoelastic model expressed in terms of the Prony series to describe the constitutive behavior of the adhesive films. The material parameters of the developed constitutive model were determined through single-step stress relaxation tests conducted for 30 min at four strain levels: 100%, 200%, 300% and 400%. Based on the reduced gradient method, the optimized material parameters were then evaluated by curve fitting the experimental data. To validate the proposed constitutive model, we performed the tensile tests at different strain rates from 5 × 10−4 to 5 × 10−1 s−1 and the multistep stress relaxation tests on the adhesive films. The model predictions and experimental data were in good agreement. Thus, the proposed hyper-viscoelastic constitutive model with parameters determined through single-step stress relaxation tests is effective in characterizing the mechanical behavior of adhesive films.

Comparison of the Trueness of Fits of the Biphasic Transverse Isotropic and Kelvin Models to the Tensile Behavior of Temporomandibular Joint Disc.

This technical brief explores the validity and trueness of fit for using the transverse isotropic biphasic and Kelvin models (first and second order generalized) for characterization of the viscoelastic tensile properties of the temporomandibular joint (TMJ) discs from pigs and goats at a strain rate of 10 mm/min. We performed incremental stress-relaxation tests from 0 to 12% strain, in 4% strain steps on pig TMJ disc samples. In addition, to compare the outcomes of these models between species, we also performed a single-step stress-relaxation test of 10% strain. The transverse isotropic biphasic model yielded reliable fits in reference to the least root mean squared error method only at low strain, while the Kelvin models yielded good fits at both low and high strain, with the second order generalized Kelvin model yielding the best fit. When comparing pig to goat TMJ disc in 10% strain stress-relaxation test, unlike the other two Kelvin models, the transverse isotropic model did not fit well for this larger step. In conclusion, the second order Kelvin model showed the best fits to the experimental data of both species. The transverse isotropic biphasic model did not fit well with the experimental data, although better at low strain, suggesting that the assumption of water flow only applies while uncrimping the collagen fibers. Thus, it is likely that the permeability from the biphasic model is not truly representative, and other biphasic models, such as the poroviscoelastic model, would likely yield more meaningful outputs and should be explored in future works.

Experimental and numerical studies on the effect of elongation rate and temperature on the mechanical behaviour of high strength NBR

ABSTRACTThis paper presents rate dependent uniaxial tensile behaviour and relaxation response of high strength acrylonitrile butadiene rubber (NBR) with moderate carbon black content. The rate dependent uniaxial tensile response is investigated for large deformations over moderate range of elongation rates (50–500 mm min−1). Effects of elongation rate on the overall deformation behaviour in terms of stress–strain response, total elongation at break as well as on the breaking stress of the material are investigated. Experimental results revealed that the material behaviour, including elongation at break and breaking stress, is rate dependent. Relaxation behaviour over a temperature range of 30–90 C, investigated using uniaxial compression experiments, shows drop of stress over the duration of the tests. Additionally, a generic rate dependent hyperviscoelastic model to predict the rate dependent response of the material and simulations of the relaxation tests are presented. Single-step stress relaxation tests from an inbuilt experimental setup and tensile response obtained from test conducted at 200 mm/min elongation rate were used in calibration. The numerical results obtained are seen to be in good agreement with the experimental observations for the range of deformation rate and temperature considered in the uniaxial tensile and relaxation tests, respectively.

Self-Healing Fish Gelatin/Sodium Montmorillonite Biohybrid Coacervates: Structural and Rheological Characterization

Complex coacervation driven by associative electrostatic interactions was studied in mixtures of exfoliated sodium-montmorillonite (Na(+)-MMT) nanoplatelets and fish gelatin, at a specific mixing ratio and room temperature. Structural and viscoelastic properties of the coacervate phase were investigated as a function of pH by means of different complementary techniques. Independent of the technique used, the results consistently showed that there is an optimum pH value at which the coacervate phase shows the tightest structure with highest elasticity. The solid-like coacervates showed an obvious shear-thinning behavior and network fracture but immediately recovered back into their original elastic character upon removal of the shear strain. The nonlinear mechanical response characterized by single step stress relaxation experiments revealed the same trend for the yield stress and isochronal shear modulus of the coacervates as a function of pH with a maximum at pH 3.0 and lower values at 2.5 and 3.5 pHs, followed by a very sharp drop at pH 4.0. Finally, small-angle X-ray scattering (SAXS) data confirmed that at pHs lower than 4.0 the coacervate phases were dense and structured with a characteristic length scale (ξ(SAXS)) of ~7-9 nm. Comparing the ξ(SAXS) with rheological characteristic length (ξ(rheol)) estimated from low-frequency linear viscoelastic data and network theory, it was concluded that both the strength of the electrostatic interactions and the conformation of the gelatin chains before and during of the coacervation process are responsible for the structure and rigidity of the coacervates.

Soft colloidal matter: A phenomenological comparison of the aging and mechanical responses with those of molecular glasses

Highly concentrated colloidal suspensions are often considered to exhibit behavior similar to that of glass-forming systems. While there is considerable rheological information in the literature concerning the flow behavior of such systems, there is little that has examined the mechanical response in a fashion that makes explicit comparisons with the relaxation behavior of molecular or polymeric glasses. On the other hand there is a significant literature that looks at “shear melting” and subsequent aging of such glasslike or “pasty” liquids. Here we report results for a polymer latex particle system at different concentrations near to the glassy or pasty regimes. Stress relaxation experiments and aging after shear melting experiments were performed. Single step stress relaxation results presented as isochrones of stress vs strain show behavior similar to that of polymers at the lower concentrations (50% and 53%). That is, there is a linear regime of behavior (generally less than a deformation of 1%) foll...

The Effect of Overshooting the Target Strain on Estimating Viscoelastic Properties From Stress Relaxation Experiments

Tendon's mechanical behaviors have frequently been quantified using the quasi-linear viscoelastic (QLV) model. The QLV parameters are typically estimated by fitting the model to a single-step stress relaxation experiment. Unfortunately, overshoot of the target strain occurs to some degree in most experiments. This has never been formally investigated even though failing to measure, minimize, or compensate for overshoot may cause large errors in the estimation of parameters. Therefore, the objective of this study was to investigate the effect of overshoot on the estimation of QLV parameters. A simulated experiment was first performed to quantify the effect of different amounts of overshoot on the estimated QLV parameters. Experimental data from tendon was then used to determine if the errors associated with overshoot could be reduced when a direct fit is used (i.e., the actual strain history was used in the curve fit). We found that both the elastic and viscous QLV parameters were incorrectly estimated if overshoot was not properly accounted for in the fit. Furthermore, the errors associated with overshoot were partially reduced when overshoot was accounted for using a direct fit. A slow ramp rate is recommended to limit the amount of overshoot and a direct fit is recommended to limit the errors associated with overshoot, although other approaches such as adjusting the control system to limit overshoot could also be utilized.

Stress relaxation of a main-chain, smectic, polydomain liquid crystalline elastomer

It has recently been shown that liquid crystalline elastomers (LCEs) exhibit bulk macroscopic orientation or a `polydomain-to-monodomain' transition when stretched in uniaxial tension (see for example review articles by: Gleim, W. and Finkelmann, H., Side-Chain Liquid Crystal Polymers, ed. C. B. McArdle. Blackie and Sons Ltd., 1989, p. 287; Zentel, R., Agnew Chem. Adv. Mater., 1989, 101(10), 1437; Barclay, G. G. and Ober, C. K., Prog. Polym. Sci., 1993, 18, 899–945). In order to investigate this phenomenon further, single-step stress relaxation experiments were performed in uniaxial tension on a polydomain, smectic LCE at variable strains relative to the polydomain-to-monodomain transition. It was found that the smectic LCE exhibited a large amount of stress relaxation and could be described at intermediate times (≈7–3000 s) by a stretched exponential function with a relatively fast characteristic relaxation time (≈60 s), regardless of the magnitude of the strain. One possible origin of this phenomenon is that the local smectic LC order is transiently disrupted during initial deformation and reorientation of the LC domains. The free energy penalty for this disruption may provide a driving force for reversion back to the original (undeformed) state of order and, correspondingly, a large amount of local chain relaxation. It was also found that the stress relaxation data were divided into two regimes. At low strains (prior to the polydomain-to-monodomain transition) the data exhibited a final relaxed modulus of E f=1.8 MPa. Samples stretched to larger strains (greater than the polydomain-to-monodomain transition) were shifted to lower values with E f=0.7 MPa.

Prediction of the subyield extension and compression responses of glassy polycarbonate from torsional measurements

Modeling of the response of solidlike polymers is often difficult, not only due to the highly nonlinear behavior of the materials but also because of the difficulty of obtaining relevant material data in the laboratory. Here, we examine the possibility of using concepts from finite elasticity theory to describe the isochronal single-step stress relaxation response for a polymer glass (polycarbonate) far below its glass transition. Torque and normal force measurements from torsional stress relaxation experiments are used to obtain isochoric values for the derivatives W1 and W2 of the strain energy density function in terms of the deformation invariants at specific time values (isochrones). The values of W1 and W2 are then used to determine isochronal values of the Valanis–Landel [Valanis, K. C. and R. F. Landel “The Strain–Energy Function of a Hyperelastic Material in Terms of the Extension Ratios,” J. Appl. Phys. 38, 2997–3002 (1967).] (VL) function derivatives w′(λ) and to predict the tension and compression responses for different deformations λ below yield. It is found that, for the conditions examined, the experimentally obtained tension and compression responses are well described within the VL framework, despite the fact that polycarbonate is a compressible material. This success suggests that the set of experiments required to describe the nonlinear behavior of glassy materials may be smaller than previously thought. Also, volumetric measurements in the uniaxial deformations indicate a densification of the glass at large deformations and long relaxation times, which is consistent with concepts in the literature that invoke mechanically accelerated aging to describe mechanical and structural interactions in the physical aging of glassy polymers.

Non-linear viscoelastic performance of Nomex, Kevlar and polypropylene fibres in a single step stress relaxation test: 2. Moduli, viscosities and isochronal stress/strain curves

The investigation of the single step stress relaxation behaviour of three polymer fibres, namely Nomex, Kevlar and polypropylene, differing widely in their mechanical properties, shows that a consistent systemization can be applied on the basis of a two-component (TC) model. The model comprises a relaxation function, based on the cumulative log-normal distribution, with a shape that is independent of the type of material and of the strain level. The limits of linearity for viscoelastic behaviour in stress relaxation are deduced and the specific sources of non-linear effects are discussed. For each of the fibres the applicability of simpler models, related to thermorheologically simple or complex materials, is discussed. The results deduced for the stresses and moduli of the elastic and viscous components of the model are plausibly explained on the basis of current models for the fibre structures. The changes of the mean relaxation times with strain are consistent with a solid/fluid transition in an amorphous ‘matrix’ component initiated at the yield point for Nomex and polypropylene fibres and with the more ‘metal-like’ nature of Kevlar.

Stress relaxation and time/temperature superposition of polypropylene fibres

The viscoelastic properties of polypropylene fibres are investigated in a single-step stress relaxation test for small strains at temperatures between −50 and + 30 °C, thus well encompassing the glass transition temperature. For the analysis of the curves a formal two-component (TC) model is applied, which comprises an elastic and a viscoelastic component, the time-dependence of which is described by a relaxation function on the basis of a cumulative log-normal distribution; the shape of this was found to be independent of temperature. The component moduli, as well as the characteristic relaxation time, show pronounced temperature-dependencies. Above the transition temperature, at ca. −15 °C, they all follow Arrhenius relationships, with activation energies which lie between −10 and −30 kJ mol−1. The consistency of the analytical model is checked against the results of dynamic extensional tests, which furthermore enable us to identify the process as the α-transition in isotactic polypropylene. Evaluation of the properties of the components of the TC model in relation to the polymer morphology suggests that the relaxation process resides in the restricted amorphous phase of the material.

Non-linear viscoelastic performance of Nomex, Kevlar and polypropylene fibres in a single-step stress relaxation test: 1. Experimental data and principles of analysis

This study, summarized in two parts, is directed at the investigation of the non-linear performance of Nomex, Kevlar and polypropylene fibres, when undergoing stress relaxation tests. The description of the strain-dependent performance is based on a two-component model, which comprises an elastic and a viscous contribution. A shape-invariant relaxation function is introduced that is based on the cumulative log-normal distribution. This first part is directed at the description of the experimental procedures and of the approach to the analysis of the stress relaxation data.

The time dependent strain potential function for a polymeric glass

Torsion and normal force measurements were made during single step stress relaxation experiments on a polymeric glass (PMMA). Isochronal data were analysed using an approach adapted from that developed by Penn and Kearsley 1 (for incompressible elastic materials) to determine the derivatives ∂W ∂I 1 , and ∂W ∂I 2 of the time dependent strain potential function. ∂W ∂I 1 and ∂W ∂I 2 are determined from existing solution to the torsion of an incompressible cylinder. A special solution to the torsion of a compressible cylinder is presented and it is shown that the values of ∂W ∂I 1 and ∂W ∂I 2 obtained using this solution to analyse the data do not differ greatly from those obtained using the incompressible solution. It is found from both solutions that ∂W ∂I 1 is negative and increases towards zero with increasing time and deformation while ∂W ∂I 2 is positive, greater in magnitude than ∂W ∂I 1 and decreases towards zero with increasing time and deformation. These results were unexpected and a full understanding of their meaning has yet to be reached.

Mechanical behaviour of isotactic polypropylene subjected to various strain histories in uniaxial extension

The mechanical behaviour of a slowly quenched isotactic polypropylene has been studied for various strain histories in extension. Creep, constant rate of strain, and constant rate of loading experiments were carried out at deformations up to and beyond the point where necking occurs. A creep diagram, which includes the failure envelope, is presented. From the available data we have also obtained an extrapolated surface of the single step stress-relaxation behaviour. From this surface we can calculate, using the Bernstein and Zapas theory on the instability of viscoelastic bars, the deformation at which necking occurs for various strain histories.

Response of Carbon Black Filled Butyl Rubber to Cyclic Loading

Abstract The stress softening behavior of a carbon black filled butyl rubber in cyclic loading (creep) and cyclic deformation (stress relaxation) histories was investigated. Single step and multiple step stress relaxation and creep experiments were conducted as were creep experiments with sinusoidal loading histories. The data were analyzed in view of the BKZ constitutive equation. For materials which follow the behavior predicted by the BKZ theory, the single step stress relaxation response is a lower bound for the stress in cyclic deformation histories, and the single step creep response is an upper bound for the deformations obtained in cyclic loading histories. In our experiments we observed that for filled butyl rubber, the responses to cyclic deformation and cyclic loading histories were outside these bounds. The softening was much more emphatic in cyclic loading histories than in cyclic deformation histories. We also found that the rubber is stiffer in creep than predicted from the BKZ theory (from stress relaxation data). It is suggested that an adequate description of material behavior might be obtained using a theory similar to the BKZ theory, which includes competing effects of stiffening in creep and softening under load-unload cycling. The failure lifetime of the filled butyl rubber in cyclic loading histories was related to the stress softening behavior. If it is assumed that, in some sense, a failure envelope exists, then it can be shown that under cyclic loading the filled butyl rubber should fail at a time earlier than under constant load conditions. We were able to predict failure lifetimes by using creep data obtained from short time cyclic loading histories and extrapolating to the strain at break determined from a shifted failure envelope. This approach relates the observed frequency and waveform dependence of failure lifetimes to the creep behavior under cyclic loading conditions.

Nonlinear Behavior of Polyisobutylene Solutions as a Function of Concentration

The nonlinear behavior in simple shear of polyisobutylene solutions in cetane was studied for three concentrations in various shear histories. The concentrations of the solutions were 10, 15.1, and 19.3% by weight. Measurements of the shear stress and the first normal stress difference were obtained as a function of time at shear strain histories involving suddenly applied shear and stress relaxation after steady shear. Steady state values were also obtained for the shear stress and first normal stress difference as a function of shear rate, and the dynamic viscosity, η′(ω), and rigidity, G′(ω) were obtained as a function of frequency. Some isochronal data is also presented from single step stress relaxation experiments. It is shown that a simple superposition principle can be applied at concentrations where the intermolecular forces are predominant for the systems and for which the potential function in the Bernstein, Kearsley , and Zapas elastic fluid theory can be expressed as a product of a function of time and a function of strain. An excellent agreement was obtained with the experimental data, even for transient experiments where the shearing and normal stresses depend on time.

Mechanical behavior of a highly oriented polyethylene rod

AbstractDynamic mechanical and single step stress relaxation data are presented for a highly oriented linear polyethylene rod as produced by a high pressure extrusion technique. The results indicate that both major relaxation processes normally observed in unoriented polyethylene are virtually absent in torsion, whereas they are present in flexure. Stress relaxation in torsion data show a very highly nonlinear behavior when compared to similar data for an unoriented polyethylene rod.

TWO STEP STRESS RELAXATION OF MODIFIED BOLTZMANN'S SUPERPOSITION PRINCIPLE AND ITS PRACTICAL APPLICATIONS

In the present paper, an equation of nonlinear superposition principle proposed by Schapery is simplified so that the dependences of equilibrium term of relaxation modulus on strain are identical with that of transient term, and this equation is named general model (model G). From this equation, the several nonlinear constitutive equations can be derived as follows.(1) Time-strain factorized model(a) Perfect factorized model (model I)……nonlinearity depends on present strain.(b) Imperfect factorized model (model II)……nonlinearity depends on strain history.(c) Mixed factorized model (model M)……to combine model I with model II.(2) Time-strain reduced model (model III)……the strain to change time scale.(3) General model (model G)……to combine model M with model III.It is difficult to assign nonlinear viscoelastic property of a given material to one of the above five models by single step stress relaxation tests. In particular, it is emphasized for power law's materials.By two step stress relaxation tests, nonlinear characteristics of materials are classified into five models as follows:The time tN, when effects of the first strain e1, decay and hence two step relaxation stress σ2(t′) under e2(=e1+Δe) meets single step relaxation stress σ1(t′) at e2, depends on only t1 (input time of Δe) for the time-strain factorized models. For other models, tN depends on e1, e2 and t1. In each case of model I, II and M, |_??_σ1/_??_e2| is linear to e1, independent of e1, and nonlinear to e1, respectively. The another useful properties are that Δσ/e1 (Δσ=σ2-σ1) decrease with e2, e1, and both e1 and e2, respectively, for model I, 11 and M. The stress increment δσ at t1 has only linear relation to Δe for the model III, while δσ is nonlinear to all e1, e2 and t1 for the model G.The theory is tested for some polyethylenes at room temperature. The samples have nearly power law's properties in single step stress relaxation behavior. The measured values of tN have properties of the model III or G. Only when Δe>0 and e1_??_2%, the experimental curves of o2(t′) agree well with the calculated curves from model III. When e1=3% and Δe>0, the calculated values are less than the experimental values. When Δe<0, the calculated values are greater than experimental values. In this case, if nonlinearized function ae (e) is provided with property of hysteresis, the theory of model III may be improved to approach the experimental results.

Simple Shearing Flows in Polyisobutylene Solutions.

Measurements of shear stress for various shearing flow histories at 25 °C are correlated through the BKZ elastic fluid theory. The data are on 10 percent PIB solution in cetane. The histories include single step stress relaxation, suddenly applied steady shear, steady shearing flow, stress relaxation after steady shear.

Stress Relaxation in Combined Torsion-Tension

In this paper we briefly examine the form of the isothermal constitutive relation of an isotropic simple material under stress-relaxation conditions. We then compare the predictions of the Signorini form of the Bernstein, Kearsley, Zapas elastic fluid theory and the Lianis constitutive equation for single-step stress-relaxation histories and show that they are identical. Using these theories, we develop theoretical expressions for the torque and axial force in combined torsion-tension stress relaxation. Experiments were performed on samples of an SBR polymer, and the observed response is compared with theoretical predictions.

Correlation of Large Longitudinal Deformations with Different Strain Histories

Abstract In 1963 Bernstein, Kearsley, and Zapas1 presented a theory of an elastic fluid which gave the correct stress-relaxation response for a large variety of elastomeric materials, including vulcanized rubbers. A principle attractiveness of this theory is its relative simplicity; with a single integral in time, it describes the stress-strain behavior for all types of deformation histories. In the case of simple extension, it predicts the behavior in any uniaxial strain history from the results of single step stress-relaxation experiments which cover the same range of extension and time. We designed a series of experiments to check the validity of this theory and found, as is shown in this paper, excellent agreement with experiment in all cases. We are aware that experiments cannot prove a theory. From our results, however, we feel strongly that a single integral expression with a nonlinear integrand such as the BKZ elastic fluid equation is sufficient to describe the stress-strain behavior of elastomeric materials.